In order to derive rotational velocities we attempted to fit Gaussian
profiles to
4 HeI lines at 4026 Å,
4143 Å, 4387 Å and 4471 Å in the spectra. This was
successful for most of the objects,
although for a small number only 2 or 3 lines could be reliably fitted due to
contamination by nearby emission or absorption features.
The profile full widths at half maximum were
converted to using a fit to the 4471 Å full width half
maximum -
correlation of Slettebak et al. (1975).
Making the appropriate correction for the differing central wavelengths of
each line, the fits employed were:
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(1) |
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(2) |
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(3) |
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(4) |
26 of the objects in our sample have previously measured 's in the
compilations of Bernacca & Perinotto (1970, 1971). A comparison
of the historical
's with those we derive shows the historical
values typically
per-cent greater than our values. However
Bernacca & Perinotto (1974) state that their
's
are referenced
to the scale of Slettebak (1968) whereas our measurements are
instead referenced to Slettebak et al. (1975).
Figure 7 of that paper
shows that the new (1975) scale derives
's some 15-20 per-cent
smaller for B stars than the old (1968) scale and so the discrepancy may
simply be understood to be caused by our use of a more modern
calibration. To make this clear Fig. 7 re-plots Fig. 7
of Slettebak et al. (1975) with their standard stars marked as
squares and our sample marked as crosses. No significant difference
is apparent between the two distributions.
In Fig. 8 (hollow plus filled areas)
we plot the distribution of versus spectral
type within the sample for luminosity classes III, IV and V.
It is important to identify whether there are are biases
in the
values within the sample. There are two effects
that may lead to a relationship between
and brightness
for Be stars and in a flux limited sample
we would of course expect an
inherent bias towards intrinsically brighter objects.
The first effect that would lead to a bias towards rapidly rotating Be stars is described by Zorec & Briot (1997). The more rapidly rotating stars will suffer greater deformation than the slower rotators. Although the bolometric luminosity from an object is of course conserved, the effect of rotation is to make the spectrum appear cooler. Since the peak of B star spectra is in the UV, this will make the optical flux brighter for most aspect ratios (Collins et al. 1991; Porter 1995), leading to the preferential selection of more rapidly rotating stars in a magnitude limited sample.
In addition it is well known that the emission produced in the circumstellar envelope of Be stars leads to an increase in their optical flux. Observations of phase changes from non-Be to Be typically show increases of 0.1-0.2 magnitudes (Feinstein 1975; Apparao 1991). This is due to reprocessing of the UV radiation from the underlying Be star into optical and infrared light by the disk. Assuming a relation between disk size and excess optical flux we would therefore expect stars with larger circumstellar disks to be preferentially selected by our flux limited sample. Also assuming that the sizes of Be star disks are likely to be sensitive to the stellar rotational velocity, then we would expect the most rapidly rotating stars to show the strongest optical excess. This again would lead to a bias towards rapidly rotating Be stars in a magnitude limited sample.
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Figure 8:
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Figure 9: Apparent B magnitude versus B spectral subclass for the volume limited sample. Symbols as per Fig. 6 |
In order to test whether our sample is biased in this way
we must compare it to a volume limited subset. This can be
created from our sample by
using the absolute magnitudes (Schmidt-Kaler 1982)
derived
from the spectral and luminosity classes to select objects
that lie within the volume defined by the absolute
magnitude limit for the intrinsically faintest
objects (B9V) at the apparent magnitude limit ().
The resulting volume limited subsample contains 34 objects out
of our original 58. We plot the B magnitude distribution of this
subsample in Fig. 9. Note how the volume limiting naturally cuts
out the objects with faint apparent magnitudes at early spectral
types (cf. Fig. 6).
The
distribution of the volume limited subsample
is plotted in Fig. 8 (filled area only) to allow comparison
with the total sample (filled plus hollow area).
In order to
compare the distributions we use a Kolmogorov-Smirnov (KS)
test between the volume-limited and total samples. This shows that
the probability of the distribution of
two samples being
the same is 80% for luminosity class III, 99% for luminosity class IV,
and 95% for luminosity class V. There is therefore no
statistical evidence for any bias in the
values for
all three luminosity classes in the sample. What is clear from
Fig. 8 is that there is a considerably
lower mean
for luminosity class III as opposed to
class V Be stars. The astrophysical interpretation of
this result is discussed in Steele (1999).
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